Microscopic Variables Research Articles

An alternative approach to microplane theory

In this paper an alternative approach to the well-known microplane theory is developed. This approach mainly consists of two parts, i.e. the mathematical representation based on the improved representation theorem on stiffness and the equivalent thermodynamical description within the framework of irreversible thermodynamics. On one hand, the material stiffness is represented in the form of irreducible decomposition which can be sufficiently determined by the orientation distribution functions for the macroscopic bulk and shear moduli (or those for the corresponding macroscopic damages). The introduced macroscopic orientation distribution functions are then expanded into the converged Fourier series and approximated by the second- or fourth-order macroscopic damage variables which are defined as the fabric tensorial functions of the microscopic (microplane) damage variables. The combination of the improved representation theorem on stiffness with the proposed macroscopic and microscopic damage variables yields the general forms of the microplane models with bulk-shear split and with volumetric–deviatoric–tangential split. On the other hand, the macroscopic Helmholtz free energy potentials are defined by introducing the damage effect tensors in terms of the macroscopic damage variables. The integral relation between the microscopic and macroscopic Helmholtz free energy potentials as well as the kinematic constraint is derived. Within the framework of irreversible thermodynamics, the consistent microscopic and macroscopic damage evolution laws are established. Moreover, the other concepts of damage mechanics such as the conjugated damage forces, the damage dissipations and so on, are also investigated on both the microscopic and macroscopic levels.

Postnatal sexual development of testis and epididymis in the rabbit: Growth and maturity patterns of macroscopic and microscopic markers

We examined the macroscopic variables related to the size of testis and epididymis, and the microscopic variables related to the tissue composition of testis to determine the onset of the male reproductive activity. The present work was carried out using two genetic lines of rabbits showing different reproductive aptitudes to assess the effects of genetic line and birth season on age-related changes of the testes and epididymis. The Caldes and Prat genetic lines showed similar developmental profiles for most of the variables studied. The main changes in the development pattern were observed at younger ages. The Caldes genetic line presented a greater live weight and a smaller testicular volume that the Prat genetic line at any age. No differences in the studied microscopic variables were found between the two genetic lines, except in the variable percentage of seminiferous tubules with presence of lumen. A significant effect of the birth season was found in live weight, testis volume, epididymis volume, percentage of seminiferous tubules with presence of elongated spermatids and diameter of seminiferous tubules. The absolute values and the values relatives to its own value at the adult stage of the variables live weight, testis volume, epididymis volume and in variables related to the functional maturity were lower in animals born in the summer season. Volume growth for both testis and epididymis was delayed in animals born in the summer season.

Homogenization of Non-linear Scalar Conservation Laws

We study the limit as ε → 0 of the entropy solutions of the equation \({\partial_t u^\varepsilon + {\rm div}_x \left[A \left(\frac{x}{\varepsilon},u^\varepsilon \right)\right] =0}\) . We prove that the sequence uε two-scale converges toward a function u(t, x, y), and u is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence result in \({L_{\rm loc}^1}\) .

Estimation of microscopic averages from metadynamics

With the help of metadynamics it is possible to calculate efficiently the free energy of systems displaying high energy barriers as a function of few selected collective variables. In doing this, the contribution of all the other degrees of freedom (microscopic variables) is averaged out and, thus, lost. In the following, it is shown that it is possible to calculate the thermal average of these microscopic degrees of freedom during the metadynamics, not loosing this piece of information.

Scale transition and enforcement of RVE boundary conditions in second‐order computational homogenization

AbstractFormulation of the scale transition equations coupling the microscopic and macroscopic variables in the second‐order computational homogenization of heterogeneous materials and the enforcement of generalized boundary conditions for the representative volume element (RVE) are considered. The proposed formulation builds on current approaches by allowing any type of RVE boundary conditions (e.g. displacement, traction, periodic) and arbitrary shapes of RVE to be applied in a unified manner. The formulation offers a useful geometric interpretation for the assumptions associated with the microstructural displacement fluctuation field within the RVE, which is here extended to second‐order computational homogenization. A unified approach to the enforcement of the boundary conditions has been undertaken using multiple constraint projection matrices. The results of an illustrative shear layer model problem indicate that the displacement and traction RVE boundary conditions provide the upper and lower bounds of the response determined via second‐order computational homogenization, and the solution associated with the periodic RVE boundary conditions lies between them. Copyright © 2007 John Wiley & Sons, Ltd.

In situ mechanical characterization of polymers with the association of three optical techniques

A combination of three optical techniques is presented for in situ monitoring of macroscopic and microscopic variables characterizing the deformation of polymers. A video extensometer allows for the monitoring of true stress and strain. An infrared imager along with an appropriate mathematical algorithm allows for the determination of the energies that are converted into heat during the whole test. This makes possible a quantification of thermomechanical couplings revealing structural effects at microscopic scales. Finally, incoherent steady light transport is used to produce images of backscattered intensities. A physical model enables the authors to follow damaging processes occurring at microscopic scales.

A micro-mechanical investigation of bifurcation in granular materials

In this paper, the notion of loss of sustainability of a mechanical state in a granular assembly is investigated. The vanishing of the second-order work, defined on the macroscopic scale from tensorial variables, is shown to play a fundamental role in detecting the occurrence of this type of bifurcation. Then a link is established between the macroscopic second-order work on the specimen scale and a discrete local expression that introduces microscopic variables defined on each contact scale. This relation opens up a micro-mechanical interpretation allowing one to examine which micro-structural features are responsible for the vanishing of the macroscopic second-order work. Finally, it is established that both geometrical and material micro-structural origins may combine to induce the occurrence of bifurcation on a specimen scale.

Asymptotics for the eigenelements of the Neumann spectral problem with concentrated masses

We consider a spectral Neumann problem for the Laplace operator posed in a domain Ω of R 3 . We assume that the density function takes the value e -m in the small region EB C Q and the value 1 outside. e is a small parameter, e ∈ (0,1), and m is a strictly positive parameter; eB is the concentrated mass. We study the asymptotic behavior, as e → 0, of the eigenvalues and the corresponding eigenfunctions for m > 2. Low and high frequencies are considered and additional information on the structure of the associated eigenfunctions is provided. We also consider the case of several concentrated masses inside the domain Q, in which for m ≥ 3 the limit problem for the low frequencies is a non-local system of equations in the microscopic variables, involving simultaneously all the domains in which the concentrated masses are placed. This strongly differs from the case where a Dirichlet condition is imposed on ∂Ω since the associated eigenfunctions lose in some way the local character affecting the concentrated mass.

Seamless multiscale modeling of complex fluids using fiber bundle dynamics

We present a seamless multiscale model and an efficient coupling scheme for the study of complex fluids. The multiscale model consists of macroscale conservation laws for mass and momentum, molecular dynamics on fiber bundles, as well as the Irving-Kirkwood formula which links the macroscale stress tensor with the microscopic variables. The macroscale and microscale models are solved with a macro time step and a micro time step respectively. The two models are synchronized at each time step by exchanging the velocity gradient (from macro to micro) and the stress tensor (from micro to macro). The multiscale method is applied to study the dynamics of polymer fluids in a channel driven by external forces.

Multiscale approach to the behaviour and damage of the heterogeneous elastic–viscoplastic materials

Both natural and man-made materials, often exhibit heterogeneous structure, depending on the scale of observation. This heterogeneous aspect acts as an important factor on their macroscopic mechanical behaviour response. One of the most reliable ways to obtain a quantitative relation between different scales is the use of the homogenisation methods. The fundamental assumption in these methods is the existence of a unit cell that is representative for the microstructure of the material under consideration, the so-called representative volume element (RVE). Modelling the mechanical behaviour of heterogeneous elastic–viscoplastic solids under finite strains framework, by using numerical homogenisation methodology is the main goal purchased in this work. Generally, the macroscopic quantities are formulated as average values of the corresponding microscopic state variables. The average of a quantity is taken over the region occupied by the unit cell [R. Hill, Elastic properties of reinforced solids: some theoretical principles, J. Mech. Phys. Solids 11 (1963) 357–372]. The method which is proposed here, has been validated by comparing results of F.E. simulations with the experimental results. The composite elements consist of a polymeric continuous matrix of polymethylmethacrylate filled with rubber particles, to obtain the so-called RT-PMMA (rubber toughened PMMA). An hyper-elastic–viscoplastic description was developed to account for both finite strains and time and history dependence mechanical behaviour that exhibits most of polymers. With regard to the influence of the damage in such kinds of composites materials, the evolution of the cavitation mechanism, identified as one of the possible damage process, is also examined.

Assessing Crash Occurrence on Urban Freeways by Applying a System of Interrelated Equations

Most existing freeway crash frequency models analyze overall frequency of crashes. Furthermore, researchers have traditionally used average annual daily traffic (AADT) to represent traffic volume in their models. These two cases are examples of macroscopic crash frequency modeling. Segregating crashes on the basis of type of crash, peak or off-peak traffic conditions, lighting conditions, severity, and pavement condition could provide insight into the specific factors that affect each category. In this study multiple binary categorizations of the crashes were created to identify the factors associated with their frequencies and used geometric characteristics of the freeway and microscopic traffic variables that were based on loop detector data. These categorizations included multiple- and single-vehicle crashes, peak period and off peak period crashes, dry and wet pavement crashes, daytime and dark-hour crashes, and property-damage-only and injury crashes. Models for frequency of each of the two groups of crashes were estimated separately for all five categorizations. To account for correlation between the disturbance terms arising from omitted variables between any two models in a category, seemingly unrelated negative binomial (SUNB) regression was used for simultaneous estimation. SUNB estimation proved to be advantageous for multiple- and single-vehicle crashes and for daytime and dark-hour crashes. Road curvature and presence of on- or off-ramps were found to be the significant factors related to every crash category. Median type and pavement surface type were among other important factors affecting crashes. AADT was significant in most models and the 15-min coefficient of variation of speed was significant for frequency of daytime and peak period crashes. SUNB estimation proved to increase the efficiency of the crash frequency models by accounting for the disturbance correlation, reducing the standard errors, and providing better model fit.

Assessing Crash Occurrence on Urban Freeways by Applying a System of Interrelated Equations

Most existing freeway crash frequency models analyze overall frequency of crashes. Furthermore, researchers have traditionally used average annual daily traffic (AADT) to represent traffic volume in their models. These two cases are examples of macroscopic crash frequency modeling. Segregating crashes on the basis of type of crash, peak or off-peak traffic conditions, lighting conditions, severity, and pavement condition could provide insight into the specific factors that affect each category. In this study multiple binary categorizations of the crashes were created to identify the factors associated with their frequencies and used geometric characteristics of the freeway and microscopic traffic variables that were based on loop detector data. These categorizations included multiple- and single-vehicle crashes, peak period and off peak period crashes, dry and wet pavement crashes, daytime and dark-hour crashes, and property-damage-only and injury crashes. Models for frequency of each of the two groups of crashes were estimated separately for all five categorizations. To account for correlation between the disturbance terms arising from omitted variables between any two models in a category, seemingly unrelated negative binomial (SUNB) regression was used for simultaneous estimation. SUNB estimation proved to be advantageous for multiple- and single-vehicle crashes and for daytime and dark-hour crashes. Road curvature and presence of on- or off-ramps were found to be the significant factors related to every crash category. Median type and pavement surface type were among other important factors affecting crashes. AADT was significant in most models and the 15-min coefficient of variation of speed was significant for frequency of daytime and peak period crashes. SUNB estimation proved to increase the efficiency of the crash frequency models by accounting for the disturbance correlation, reducing the standard errors, and providing better model fit.

Strain gradient plasticity by internal-variable approach with normality structure

This paper presents a constitutive formulation for materials with strain gradient effects by internal-variable approach with normality structure. Specific micro-structural rearrangements are assumed to account for the inelasticity deformations for this class of materials, and enter the constitutive formulations in form of internal variables. It is further assumed that the kinetic evolution of any specific micro-structural rearrangement may be fully determined by the thermodynamic forces associated with that micro-structural rearrangement, by normality relations via a flow potential. Macroscopic gradient-enhanced inelastic behaviours may then be predicted in terms of the microscopic internal variables and their conjugate forces, and thus a micro–macro bridging formulation is available for strain-gradient-characterised materials. The obtained formulations are first applied to crystallographic materials, and a crystal gradient plasticity model is developed to account for the influence of microscopic slip rearrangements on the macroscopic gradient-dependent mechanical behaviour for this class of materials. Micro-cracked geomaterials are also treated with these formulations and a gradient-enhanced damage constitutive model is developed to address the impacts of the evolutions of micro-cracks on the macroscopic inelastic deformations with strain gradient effects for these materials. The available formulations are further compared with other thermodynamic approaches of constitutive developing.

Recovery of duodenal villi and cells in chickens refed protein, carbohydrate and fat

1. To clarify how histological recovery of villi and cells would be affected after refeeding single nutrients such as protein, carbohydrate and fat, male chickens were divided as follows: (1) intact control fed ad libitum a commercial finisher mash diet (CP, 140 g; ME, 11·71 MJ/kg, ALM), (2) 3 d feed withdrawal (FW), (3) FW followed by one day ad libitum free access to the mash diet (FW-ALM), and FW followed by one day force-feeding of (4) a commercial finisher pellet diet (FW-FFM) and an isocaloric diet of (5) a protein (FW-FFP), (6) a carbohydrate (FW-FFC) or (7) a fat (FW-FFF). 2. After refeeding, the formula diet groups increased in villus height and villus area and tended to increase in cell area and cell mitosis. Furthermore, flat cells on the villus tip in the F group developed to dome-shaped cells. This suggests that nutritionally well-balanced diets can induce histological recovery at villus and cellular levels. 3. Not all of the single nutrient groups recovered to the extent of the formula diet groups in all light microscopic variables after refeeding, suggesting that a single nutrient cannot induce histological recovery of the villus. 4. However, the dome-shaped cells were more distributed on the villus tip in these single nutrient groups than in the well-balanced formula diet groups, although cell diameter of the former groups was smaller than that of the latter. This suggests that the single nutrients would be effectively absorbed from cells and can induce histological recovery at the cellular level.

Kinematic variables bridging discrete and continuum granular mechanics

It is known that there is wide, and at present, unbridgeable, gap between discrete and continuum granular mechanics. In this contribution, first, microscopic kinematic variables neglected in classical continuum granular mechanics are investigated based on the kinematics of discs in contact. Then, a kinematic variable called the averaged pure rotation rate (APR) is proposed for an assembly of circular discs of different sizes, which is then used to produce another two kinematic tensors with one equal to the deformation rate tensor and the other unifying the spin tensor and the APR. As an example, the kinematic variables are incorporated into the unified double-slip plasticity model. Finally, these theoretical analyses are verified using a two-dimensional discrete element method. The study shows that these kinematic variables can be used to bridge discrete and continuum granular mechanics.

Estimation of CE-CVM energy parameters from miscibility gap data

The powerful framework of cluster expansion- cluster variation methods (CE- CVM) expresses alloy free energy in terms of energy (model) parameters, macroscopic variables (composition and temperature) and microscopic variables (correlation functions). A simultaneous optimization of thermodynamic and phase equilibria data using CE- CVM is critically dependent on giving good initial values of energy parameters, macroscopic and microscopic variables, respectively. No standard method for obtaining the initial values of the energy parameters is available in literature. As a starting point, a method has been devised to estimate the values of energy parameters from consolute point (miscibility gap maximum) data. Empirical relations among energy parameters, temperature (T c), composition (x c) andd 2 T/dx2 at the consolute point, have been developed using CE- CVM free energy functions for bcc and fcc structures in the tetrahedron and tetrahedronoctahedron approximations, respectively. Thus from the observed data ofT c,x c andd 2 T/dx2 in the above relations, good initial values of energy parameters can be obtained. Further, a necessary modification to the classical NR method for solving simultaneous nonlinear/transcendental equations with a double root in one variable and a simple root in the other has been presented.

Steady-state granular flow in a 3D cylindrical hopper with flat bottom: macroscopic analysis

The information of a hopper flow at a particle scale, obtained from discrete particle simulation, is used to investigate the macroscopic dynamic behaviour of granular flow in a cylindrical hopper with flat bottom by means of an averaging technique. The macroscopic properties including velocity, mass density, stress and couple stress are quantified under the cylindrical coordinate framework, and an effort is made to link these variables to the microscopic variables considered. The velocity and density distributions are first illustrated to match qualitatively the experimental and numerical results, confirming the validity of the proposed averaging method. Four components of stress, T zz , T rr , T rz and T zr , and two dominant components of couple stress, M r θ and M z θ , are then investigated in detail. It is shown that large vertical normal stress is mainly observed in the region close to the bottom corner, large radial normal stress is observed within the particle bed as well as the bottom corner, and large shear stresses in the region adjacent to the vertical wall. The four stresses are relatively small in a region close to the orifice. Their magnitudes are mainly contributed by the interaction forces between particles and between particles and walls. However, the transport of particles also plays a significant role at the orifice, especially, in the vertical normal stress. The couple stress can be ignored except for the regions close to the vertical and bottom walls, where the most dominant components are M r θ adjacent to the vertical wall and M z θ close to the bottom wall. The magnitudes of these macroscopic variables depend on the geometric and physical parameters of the hopper and particles such as the orifice size and wall roughness of the hopper, and the friction and damping coefficients between particles although their spatial distributions are similar.

Semi-empirical derivation of the physical approximants to a-CN:H film deposition

An extremely simplified picture of the film deposition process is demonstrated by a semi-empirical application of the Π theorem of dimensional analysis. The growth of hydrogenated carbon-nitrides (a-CN:H) by reactive sputtering is considered as an example and elementarily modelled. By theoretically deriving a scaling law for the microscopic variables involved, a single dimensionless combination of the deposition parameters is generated, providing the rules according to which the growth conditions can be changed without significantly modifying the film nanostructure and stoichiometry. The physical laws governing the process are approximated through analytical functions of this combination, able to effectively account for the compositional and structural changes described by the evolution of the Raman spectra of the films. These functions, theoretically deduced from very simple models, are empirically demonstrated by fitting the experimental data. The efficacy of the proposed method, applicable to all materials and deposition techniques, is here shown for a given variation range of the growth conditions.

A continuum damage mechanics model for unidirectional composites undergoing interfacial debonding

A continuum damage mechanics (CDM) model is developed in this paper for fiber reinforced composites with interfacial debonding. The model is constructed from rigorous micromechanical analysis of the Representative Volume Element (RVE) using the Voronoi cell FEM (VCFEM) that is followed by homogenizing microscopic variables using asymptotic homogenization. The microstructural damage mode considered in this paper is fiber–matrix interfacial debonding that is simulated using cohesive zone models in VCFEM. Following a systematic consideration of various order damage tensors, an anisotropic CDM model using fourth order damage tensor with stiffness characterized as an internal variable, is found to perform most accurately for this class of materials. The comparison of this CDM results with those obtained by homogenization of micromechanical analysis show excellent agreement between the two. Hence the CDM model is deemed suitable for implementing in macroscopic finite element codes to represent damage evolution in composites with significant efficiency.

Thermodynamic relationship between creep crack growth and creep deformation

Crack growths processes of various types are widely covered by power laws. Especially, both creep deformation of creep-resistant bulk materials and creep crack growth in such materials can be covered by power laws and possess close exponents in many cases [1]. This paper focuses on the microscopic thermodynamic mechanisms of the correspondence and the power law itself. In this paper, it is shown that the power laws can be considered as certain homogenous kinetic rate laws of local or microscopic internal variables within the thermodynamic framework of Rice [2, 3] and correspond to a certain macroscopic requirement of maximum dissipation. It is revealed that nonlinear phenomenological equations and Onsager reciprocal relations emerge naturally from the framework if each rate is a monotonic increasing and homogeneous function of the same degree in its conjugate force. The homogeneity property transfers exactly from local internal variables to global internal variables. On the basis of the remarkable properties, it is shown that the power laws of crack growth directly lead to the refined Griffith criterion by Rice [4], and both exponents of creep deformation and creep crack growth can be related by a simple linear relation.